Nonlinear finite elements/Motion in Lagrangian form

Motion in Lagrangian form

The motion is given by

${x=\varphi (X,t)=x(X,t)~,~~\qquad X\in [0,L_{0}]~.}$

For the reference configuration,

$X=\varphi (X,0)=x(X,0)~.$

The displacement is

${u(X,t)=\varphi (X,t)-X=x-X~.}$

For the reference configuration,

$u_{0}=u(X,0)=\varphi (X,0)-X=X-X=0~.$

The deformation gradient is

${F(X,t)={\frac {\partial }{\partial X}}[\varphi (X,t)]={\frac {\partial x}{\partial X}}~.}$

For the reference configuration,

$F_{0}=F(X,0)={\frac {\partial }{\partial X}}[\varphi (X,0)]={\frac {\partial X}{\partial X}}=1~.$

The Jacobian determinant of the motion is

${J={\cfrac {A}{A_{0}}}F~.}$

For the reference configuration,

$J_{0}={\cfrac {A_{0}}{A_{0}}}F_{0}=1~.$

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